Case Study: Real‑World Results Using ProbGee### Executive summary
ProbGee is a probabilistic programming toolkit designed to make Bayesian modeling, uncertainty quantification, and probabilistic inference more accessible to data scientists and engineers. This case study examines how ProbGee was applied at Acme Logistics (pseudonym) to improve demand forecasting, reduce stockouts, and optimize inventory holding costs. The project demonstrates measurable gains in forecast accuracy, decision confidence, and operational savings.
Background and business problem
Acme Logistics operates a network of regional warehouses supplying retail stores. The company historically used deterministic time‑series forecasting (seasonal ARIMA with point estimates) combined with static safety stock rules. Challenges included:
- Frequent stockouts during promotional periods and irregular demand spikes.
- Excess inventory during slow seasons due to conservative safety stock buffers.
- Difficulty quantifying forecast uncertainty for downstream procurement and routing decisions.
Business goals for the ProbGee pilot:
- Reduce stockout rate by at least 20% for pilot SKUs.
- Decrease average inventory holding cost by 10%.
- Provide actionable probabilistic forecasts with interpretable uncertainty intervals for planners.
Why ProbGee?
ProbGee was chosen for three main reasons:
- Flexible probabilistic modeling primitives that integrate time series, hierarchical structures, and covariates.
- Scalable inference engine (variational inference + MCMC hybrid) suitable for hundreds of SKUs.
- User‑friendly APIs and visualization tools for uncertainty communication to non‑technical stakeholders.
Key decision: use Bayesian hierarchical forecasting models in ProbGee to share strength across related SKUs and regions while capturing SKU‑specific noise.
Data and preprocessing
Dataset: 18 months of daily sales for 1,200 SKUs across 12 regions, plus calendar features (promotions, holidays), price, and store openings.
Preprocessing steps:
- Aggregated sales to weekly level to reduce noise and align with replenishment cadence.
- Encoded promotions as binary flags and as percent price discounts.
- Imputed missing weeks for new SKUs using hierarchical priors (warm start from category averages).
- Split into training (first 14 months), validation (next 2 months), and test (final 2 months).
Feature engineering examples:
- Lag features (1, 2, 4 weeks) and moving averages.
- Interaction terms between promotion flag and weekday effect.
- External demand index constructed from web traffic and social media mentions.
Model architecture
We built a hierarchical Bayesian time‑series model in ProbGee with these components:
- Global level: shared priors for baseline demand and seasonality across SKU categories.
- SKU level: SKU‑specific baseline, trend, and promotion sensitivity modeled as random effects.
- Region level: regional multipliers for baseline demand.
- Observation model: Negative Binomial likelihood to account for overdispersion in counts.
- Covariates: price elasticity, promotion flags, holiday indicators, external demand index.
- Time dynamics: local linear trend plus seasonal components (annual and weekly) modeled with Gaussian state‑space priors.
Inference method:
- Initial parameter estimates via ProbGee’s amortized variational inference for speed.
- Final posterior sampling for selected pilot SKUs using Hamiltonian Monte Carlo (HMC) to obtain high‑quality uncertainty estimates.
Model training details:
- Trained on a distributed cluster using ProbGee’s built‑in data loaders; average training time ~3 hours per model family.
- Memory and computational constraints dictated batching by category; posterior samples for all SKUs collected asynchronously.
Evaluation metrics
We evaluated models on:
- Mean Absolute Percentage Error (MAPE) for point forecasts.
- Prediction Interval Coverage Probability (PICP): proportion of observations within 80% and 95% credible intervals.
- Stockout rate: percent of replenishment periods where demand exceeded available inventory.
- Inventory holding cost: calculated as average inventory level × per‑unit holding cost.
- Expected Cost of Stockouts vs. Holding: decision‑centric metric computed using probabilistic demand forecasts and reorder policy simulations.
Baseline: existing ARIMA point‑forecast + fixed safety stock policy.
Results — accuracy and uncertainty
Point forecast accuracy:
- Baseline MAPE (pilot SKUs): 18.7%
- ProbGee hierarchical model MAPE: 13.2% (29% relative improvement)
Uncertainty calibration:
- 80% credible interval PICP: 78% (close to nominal)
- 95% credible interval PICP: 94% (well‑calibrated)
Interpretation: ProbGee produced more accurate point forecasts and well‑calibrated uncertainty intervals, enabling safer decision thresholds.
Results — operational impact
Stockouts:
- Baseline stockout rate (pilot period): 6.5%
- After implementing ProbGee‑driven reorder policy: 4.1% (37% relative reduction)
Inventory holding cost:
- Baseline average holding cost (pilot SKUs): $1.12 million over test period
- ProbGee approach: $1.01 million (10% reduction)
Total expected cost (holding + stockouts) decreased by 14%, driven by smarter safety stock levels informed by SKU‑level uncertainty rather than blunt multipliers.
Example: decision policy using probabilistic forecasts
The team replaced fixed safety stock rules with a risk‑based reorder rule:
- Compute predictive demand distribution for lead time L.
- Choose target service level α (e.g., 95%) and set reorder point to the α‑quantile of the predictive distribution.
- For high‑impact SKUs, increase α to 98% after cost‑benefit analysis.
This policy was simulated with historical lead times and resulted in the reported stockout and cost improvements.
Implementation lessons and challenges
- Data quality: inconsistent promotion tagging required manual cleaning; investing in upstream data governance paid off.
- Computational cost: full HMC for all SKUs was prohibitively expensive; using amortized VI for most SKUs and HMC for high‑value SKUs provided a good tradeoff.
- Change management: visualizing predictive intervals and expected costs helped planners trust probabilistic outputs.
- Model monitoring: set up weekly calibration checks and automatic retraining triggers when PICP drifted.
Sensitivity analysis
We performed sensitivity checks on:
- Lead‑time variability: probabilistic lead‑time modeling slightly increased reorder points but reduced unexpected stockouts.
- Promotion intensity assumptions: misspecifying promotion effect priors led to higher MAPE; using hierarchical priors mitigated this.
- SKU grouping granularity: overly broad grouping reduced SKU‑level accuracy; a mid‑level category hierarchy balanced data sharing and specificity.
ROI and business case
Over a 12‑month rollout projection for similar SKU sets:
- Projected annual savings: ~$3.2M from reduced stockouts and lower holding costs.
- Implementation cost (engineering + licensing + compute): ~$600K first year.
- Estimated payback period: ~3 months post‑deployment for pilot SKU cohort.
Conclusions
ProbGee enabled Acme Logistics to move from deterministic forecasts and blunt inventory rules to probabilistic, decision‑centric forecasting. The approach yielded notable improvements in forecast accuracy, better‑calibrated uncertainty, reduced stockouts, and lower holding costs. Key success factors were hierarchical modeling, selective use of high‑quality posterior sampling, and stakeholder visualization of uncertainty.
Appendix — technical snippets
Example ProbGee model pseudocode (conceptual):
from probgee import Model, HierarchicalPrior, NegBinomial, LocalLinearTrend with Model() as model: # Global priors mu_base = HierarchicalPrior('mu_base', group='category') sigma_base = Prior('sigma_base', LogNormal(0,1)) # SKU-level effects sku_offset = RandomEffect('sku_offset', groups='sku', prior=Normal(0, sigma_base)) # Time dynamics trend = LocalLinearTrend('trend', groups='sku') seasonality = Seasonal('season', period=52, groups='sku') # Observation lambda_ = exp(mu_base + sku_offset + trend + seasonality + beta_prom*promo + beta_price*price) obs = NegBinomial('sales', mu=lambda_, phi=phi, observed=weekly_sales) # Inference vi = model.fit(method='amortized_vi', epochs=200) posterior = model.sample_posterior(method='hmc', groups=high_value_skus, samples=1000) End.
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